Analysis of Functions I: increase, decrease, and concavity

[josh_123]

Hello I need help with these problems. The direction said

a. find the intervals on which f is increasing, b. the intervals on which f is decreasing, c. the open intervals on which f is concave up, d. the open intervals on which f is concave down and e. the x-cordincates of all inflection points
1. f(x)=x^4-8x^2+16
I find a,b c and d for this function. However I have trouble finding the inflection points for this function. When you find the inflection point you suppose to set up the second derivative of the function to equal 0 but I have no idea how to solve it afterward. Is there an inflection point for this equation?
f"(x)=12x^2-16
2. f(x)=x/(x^2+2)
so f'(x)=-x²+2/(x²+2)²
to find a,b I have to set this equal to 0 and solve it. How do I solve it? and what would be the second derivative and how do I solve it by setting it up to equal to 0 to know if it's concave up or down?
3. f(x)=x²lnx
The first derivative is 2xlnx+x. How do I solve it or know what is a, b is?and the second derivative is 2lnx+3. How do I use the second derivative to solve for c,d and e?

Please help! Thank you!

 

[ehild]

1. f(x)=x^4-8x^2+16
I find a,b c and d for this function. However I have trouble finding the inflection points for this function. When you find the inflection point you suppose to set up the second derivative of the function to equal 0 but I have no idea how to solve it afterward. Is there an inflection point for this equation?
f"(x)=12x^2-16

So you need to solve the equation 12x^2-16 = 0. What is your problem?

2. f(x)=x/(x^2+2)
so f'(x)=-x²+2/(x²+2)²
to find a,b I have to set this equal to 0 and solve it. How do I solve it? and what would be the second derivative and how do I solve it by setting it up to equal to 0 to know if it's concave up or down?

Use parentheses. The formula is wrong without them.
You will have a fraction. It can be zero if the nominator is zero.

3. f(x)=x²lnx
The first derivative is 2xlnx+x. How do I solve it or know what is a, b is?and the second derivative is 2lnx+3. How do I use the second derivative to solve for c,d and e?

To find x where the derivative is zero, factor out x.

ehild

 

[HallsofIvy]

Hello I need help with these problems. The direction said

a. find the intervals on which f is increasing, b. the intervals on which f is decreasing, c. the open intervals on which f is concave up, d. the open intervals on which f is concave down and e. the x-cordincates of all inflection points
1. f(x)=x^4-8x^2+16
I find a,b c and d for this function. However I have trouble finding the inflection points for this function. When you find the inflection point you suppose to set up the second derivative of the function to equal 0 but I have no idea how to solve it afterward. Is there an inflection point for this equation?
f"(x)=12x^2-16

What? You don't know how to solve [itex]12x^2- 16= 0[/itex]? Add 16 to both sides, divide both sides by 12, then take the square root of both sides.

2. f(x)=x/(x^2+2)
so f'(x)=-x²+2/(x²+2)²
to find a,b I have to set this equal to 0 and solve it. How do I solve it?

you are taking Calculus and do not know how to solve equations like this? Multiply both sides of the equation by that denominator to get [itex]2- x^2= 0[/itex]

and what would be the second derivative and how do I solve it by setting it up to equal to 0 to know if it's concave up or down?

Differentiate it, using the quotient rule again.

3. f(x)=x²lnx
The first derivative is 2xlnx+x. How do I solve it or know what is a, b is?

Oh, c'mon! 2xln(x)+ x= x(2ln(x)+ 1)= 0. One thing you surely learned long ago is that a product is equal to 0 only if one or more of the factors is 0. So here, either x= 0 or 2ln(x)+ 1= 0: x= 0 or ln(x)= -1/2.

and the second derivative is 2lnx+3. How do I use the second derivative to solve for c,d and e?

2ln(x)+ 3= 0 leads to ln(x)=-3/2. Again, you should have learned in algebra or precalculus that if ln(x)= a then [itex]x= e^a[/itex]. Once you know where it zero, you can determine the intervals in which the second derivative is positive or negative.

Please help! Thank you!

 

[josh_123]

Actually I finished this before checking back haha. Please delete this post